A little history of e

Some students remarked on why I actually recognise e, that is, e=2.718281828.... Well, e is a rather unique constants. Firstly, for all JC students, we see it our daily algebra & complex numbers. Students exposed to university statistics will see e appearing in the formula for normal distribution, that is, f(x | \mu , \sigma^2) = \frac{1}{\sqrt{2 \sigma^2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}.

Secondly, the story of how it came about is pretty cool as you will observed in the video below.

The Story of e

Hopefully it provides you with another perspective towards this constants! And now you should be more cautious when signing up savings plans that give interest per annum or per month.

P.S. I once confused a banker when I asked her about this. 🙂

The man who knew infinity

Previously, I shared a post on the golden nugget. It relates a bit about the movie The Man Who Knew Infinity. I finally found time to watch it while I’m overseas. Usually in Singapore, I’m too overwhelmed with work and tuitions. So yes, I actually watch movies and gym/ swim when I’m overseas. So the show is really great. Its not a MATH movie so one doesn’t have to understand any math to watch it. It traces the life story of S. Ramanujan, who if you watched the video on the golden nugget, is a seriously good mathematician. The whole movie is really exciting and tells a good story of the sacrifices that mathematicians make. And I do hold great respects for anyone in the field of Pure Mathematics. The things they do really extends to solve many real-world problem today. And if you’re interested in prime numbers a bit, feel free to read here.

Another great math-related movie recently will be the Imitation Game, which on Alan Turing. I watched it with some of my closer students and most of them find it non-mathematical.

Yes, this is another “motivating Mathematics” post.

Golden Nugget!!!

1 + 2 + 3 + 4 + 5 + ... = -\frac{1}{12}

My brother shared this interesting video with me a few days back when he was at the screening for The Man Who Knew Infinity. I’m looking forward to watching this movie too!

Back to the video! It focuses on the sum that is written above. And interestingly, this sum that should not be defined (as what JC students learnt in Arithmetic Progression), is actually a NEGATIVE number. Explain excellently by professor Edward Frenkel. He brings in interesting concepts from complex numbers too. Hopefully, this piques some interest!

Professor Edward Frenkel wrote a book “Love & Math”, which is really intriguing. You don’t have to love math to read it but you will after reading 🙂

You can read more here too.

Introduction to Stochastic Calculus

Back in undergraduate days, when I took my first module on financial mathematics, my professor introduced us by that the most important things are the following

(\Omega, \mathcal{F}, \mathcal{P})

This is a probability triple where
1. \mathcal{P} is the ‘true’ of physical probability measure
2. \Omega is the universe of possible outcomes.
3. \mathcal{F} is the set of possible events where an event is a subset of \Omega.

There is also a filtration \{\mathcal{F}_t\}_{t \ge 0}, that models the evolution of information through time. For example, if by time t, we know that event \mathcal{E} has occurs, then \mathcal{E} \in \mathcal{F}_t. In the case of a finite horizon from [0,T], then \mathcal{F} = \mathcal{F}_T

A stochastic process X_t is \mathcal{F}_t-adapted if the value of X_t is know at time t when the information represented by \mathcal{F}_t is known. Most of the times, we have sufficient information at present.

In the continuous-time model, \{\mathcal{F}_t\}_{t \ge 0} will be the filtration generated by the stochastic processes (usually a brownian motion, W_t), based on the model’s specification.

Next, we review some martingales and brownian motion, alongside with quadratic variation here.

Question of the Day #17

This is a pretty cool and interesting question from AMC.

Source: 3qdigital.com
Source: 3qdigital.com

There are four lifts in a building. Each makes three stops, which do not have to be on consecutive floors on include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum number of floors that this building can have?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 12